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        <td class="header">&nbsp; Compact Polarimetric Decomposition<br>
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<h3>Compact Polarimetric Decomposition Operator</h3>&nbsp;&nbsp;&nbsp;This
operator performs the following decompositions on compact pol data:<br>
<ul>
    <li>m-chi decomposition<br>
    </li>
    <li>m-delta decomposition</li>
    <li>H-Alpha decomposition</li>
    <li>2-layer RVOG decomposition<br>
    <li>Model-Free 3-Component Decomposition&nbsp;(MF3CC)<br>
    </li>
</ul>
&nbsp;&nbsp; The operator first computes Stokes vector [g0,
g1, g2, g3] for each pixel from the averaged covariance matrix, then
performs user selected decomposition.<br>
<h5>m-chi Decomposition:</h5>

<div style="text-align: center;">&nbsp;<img style="width: 169px; height: 124px;" alt="" src="images/m_chi_eq1.jpg"><br>
</div>
where <br>

<div style="text-align: center;"><img style="width: 169px; height: 92px;" alt="" src="images/m_chi_eq2.jpg"><br>
</div>
<h5>m-delta decomposition:</h5>

<div style="text-align: center;"><img style="width: 169px; height: 124px;" alt="" src="images/m_delta_eq1.jpg"><br>
</div>
where<br>

<div style="text-align: center;"><img style="width: 169px; height: 97px;" alt="" src="images/m_delta_eq2.jpg"><br>
</div>
<br>
<h5>H-Alpha decomposition:</h5>
<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;" lang="EN-GB"><span style="position: relative; top: 19pt;"><br>
</span></span>

<div style="text-align: center;"><img style="width: 164px; height: 60px;" alt="" src="images/h_alpha_eq1.jpg"><br>
</div>
<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;" lang="EN-GB"><span style="position: relative; top: 19pt;"></span></span><br>
The wave entropy is computed from the eigen decomposition of the 2x2 covariance matrix C<sub>2</sub>. Let
&#955;<sub>1</sub> and &#955;<sub>2</sub> be the two eigen values, the entropy is given by<br>
<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;" lang="EN-GB"><span style="position: relative; top: 6pt;"><br>
</span></span>

<div style="text-align: center;"><img style="width: 219px; height: 28px;" alt="" src="images/h_alpha_eq2.jpg"><br>
</div>
where<br>

<div style="text-align: center;"><img style="width: 148px; height: 53px;" alt="" src="images/h_alpha_eq3.jpg"><br>
</div>
The compact pol anisotropy is defined by<br>

<div style="text-align: center;"><img style="width: 148px; height: 53px;" alt="" src="images/h_alpha_eq4.jpg"><br>
</div>
<br>
<h5>RVOG decomposition:</h5>

<div style="text-align: center;"><img style="width: 182px; height: 100px;" alt="" src="images/rvog_eq1.jpg"><br>
</div>

<h4>Model-Free 3-Component Decomposition&nbsp;(MF3CC)</h4>
Model-free 3 component decomposition technique [4] does not consider any volume model for the computation of the three scattering power components. 
The scattering power components are roll-invariant. The total power is conserved after decomposition and all the scattering power components
are non-negative. The target scattering type parameter is represented as:
<center><img style="width: 300px; height: 130px;" alt="" src="images/MF3CC_parm.png"></center><br>

where, &#920<sub>CP</sub> is the target characterization parameter which varies from -45 to 45 degrees, m<sub>CP</sub> is the 2D Barakat degree of polarization,
S<sub>0</sub>, S<sub>1</sub>, S<sub>2</sub> and S<sub>3</sub> are the elements of the Stokes vector and S<sub>0</sub> is the total power of covariance
matrix C<sub>2</sub>.

<br></br>

The scattering power components are:

<center><img style="width: 300px; height: 150px;" alt="" src="images/MF3CC_powers.png"></center><br>
<br>
P<sub>d</sub><sup>CP</sup> is the even bounce power component, P<sub>s</sub><sup>CP</sup> is the odd bounce power component and 
P<sub>v</sub><sup>CP</sup> is the diffused power component.

<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;" lang="EN-GB"><span style="position: relative; top: 19pt;"><br>
</span></span>

<h4>Input and Output</h4>

<ul>
    <li>The
        input to this operator should be a simulated compact polarimetric SAR
        product in complex scattering vector or complex covariance matrix
        format.
    </li>
    <li>The output of this operator is the m-chi, m-delta, H-Alpha or RVOG decomposition result represented by RGB
        colours.
    </li>
</ul>
<ol>
</ol>
<h4>Parameters Used</h4>&nbsp;&nbsp; The following processing parameters are needed (see Figure 1):<br>
<ul>
    <li>Decomposition: m-chi, m-delta, H-Alpha, RVOG decomposition or Model-Free 3-Component Decomposition<br>
    </li>
    <li>Window Size X: X dimension of the sliding window used in computing the mean covariance matrix</li>
  <li>Window Size Y: Y dimension of the sliding window used in computing the mean covariance matrix</li>

</ul>
<br>
<img style="width: 450px; height: 450px;" alt="" src="images/CompactPolDecompositionOp.jpg"><br>

<div style="text-align: left;">&nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; <br>&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
    &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;Figure 1. Dialog box for
    Compact Polarimetric Decomposition operator<br></div>
<br><br>

<p> Reference:&nbsp;</p>

<p>[1] R. K. Raney, J. T. S. Cahill, G. W. Patterson, and B. J. Bussey,
    "The m-chi decomposition of hybrid dual-polarimetric radar data with
    application to lunar craters", Journal of Geophysical Research, Vol.
    117, E00H21, doi:10.1029/2011JE003986, 2012.<br>
</p>

<p>[2] R. K. Raney, "Hybrid-Polarity SAR Architecture", IEEE
    transaction on Geoscience and Remote Sensing, Vol. 45, No. 11, Nov,
    2007.</p>

<p>[3] S. R. Cloude, D. G. Goodenough, H. Chen, "Compact Decomposition
    Theory", IEEE Geoscience and Remote Sensing Letters, Vol. 9, No. 1,
    Jan. 2012.<br>
    <br>
    <p>[4] S. Dey, A. Bhattacharya, D. Ratha, D. Mandal and A. C. Frery, 2020. 
        &#8220;Target characterization and scattering power decomposition for 
        full and compact polarimetric SAR data&#8221;, IEEE Transactions on 
        Geoscience and Remote Sensing, 59(5), pp.3981-3998. </p>
</p>


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